Reduced order models, i.e. models that are simpler than the DFN model but retain most of its predictive capabilities, can improve physical understanding and provide accurate solutions at a significantly lower computational cost. The most popular family of reduced order models are the Single Particle Models (SPMs). The key idea of SPMs is that the behaviour of the particles within each electrode is very similar and, therefore, they can all be approximated by a single representative (or average) particle, hence the name. The foundations of the Single Particle Model were first introduced in 1979 by Atlung et al. [1]} considering only the mass transport in three different particle geometries: planar, cylindrical and spherical. Recently, these models have garnered renewed interest from the battery modelling community, with several authors presenting different versions of SPMs, with and without electrolyte dynamics. There are two main approaches to formulating these models. While some authors take an ad hoc approach, posing the model directly from a list of simplifying assumptions [2]}, [3]}, [4]}, [5]}, [6]}, others (particularly in the mathematical modelling community) use formal asymptotic methods to systematically derive the reduced order models directly from the DFN model. Over the past few years, several authors have applied asymptotic methods to the DFN model to derive reduced order battery models [7]}, [8]}, [9]}, [10]}, [11]}. The exact form of the model that results from this process depends upon the underlying assumptions on which the asymptotic analysis is based.
